b. tech. [sem vii (ce, me, it, ei, en, cs & ec – all...

EOE-073/ Mr. S. B. Tiwari, Mr. J. P. Ojha, Mr. A. K. Mishra, Dr. Akhilesh Mishra & Mr. Rupesh Srivastav. Date: 10 / 09 / 2012

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SHRI RAMSWAROOP MEMORIAL COLLEGE OF ENGG. & MANAGEMENT

B. Tech. [SEM VII (CE, ME, IT, EI, EN, CS & EC – All Groups)] QUIZ TEST-1

(Session: 2012-13)

OPERATIONS RESEARCH (EOE-073)

Roll No.

(To be filled by the Student) Note: Attempt all questions

Part – A (Questions from Tutorial Sheet)

[10 Marks]

1) A manufacture of wooden articles produces tables and chairs which requires two types of inputs mainly,

these being wood and labour. The manufacturer knows that for a table 3 units of wood and 1 unit of

labour are required while for chair these are 2 units and 2 units respectively. The profit from each table

is Rs.20 while it is Rs.16 for each chair. The total available resources for the manufacturer are 150 units

of wood and 75 units of labour. The manufacturer wants to maximize his profit by distributing his

resources for tables and chairs. Formulate this as a linear programming problem. (Q. No. 3-i) [04]

2) Use two-phase simplex method to solve the problem:

Minimize 21 32

15 xxZ −= s. t. 33 321 ≥−− xxx ; 2321 ≥+− xxx and 0,, 321 ≥xxx . (Q. No. 8) [06]

Part – B [20 Marks]

3) Use Simplex method to solve: [08]

Max. Z = 2x1 + x2 subject to x1 - x2 ≤ 10; 2x1 – x2 ≤ 40; x1, x2 ≥ 0.

4) Solve the following linear programming problem graphically: [06]

Min. 21 25 xxZ −= s. t. 132 21 ≥+ xx ; 0, 21 ≥xx .

5) What is operations research? Explain general method for solving operations research models. [02+04]

__________X__________

Academic/26 Refer/WI/ACAD/18

Time: 1 Hour Max. Marks: 30


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(Session: 2012-13)


Roll No.



[10 Marks]

1) Define Operations Research? Throw light on the growing importance of O.R. in business decisions.

(Q. No. 1) [02+02]

2) Use Simplex method to solve:

Max. Z = 3x1 + 5x2 + 4x3 Subject to 2x1 + 3x2 ≤8; 2x2 + 5x3 ≤ 10; 3x1 + 2x2 + 4x3 ≤ 15 and x1, x2, x3 ≥ 0.

(Q. No.6) [06] Part – B

[20 Marks]

3) Solve the following linear programming problem: [08]

Min Z = x1 - 2x2 - 3x3 subject to -2x1 + x2 + 3x3 = 2; 2x1 + 3x2 + 4x3 = 1 and x1, x2, x3 ≥ 0.


Min. 21 25 xxZ −= s. t. 132 21 ≥+ xx ; 0, 21 ≥xx .

5) A toy company manufactures two types of doll; a basic version – doll A and a deluxe version – doll B.

Each doll of type B, takes twice as long to produce as one of type A, and the company would have time

to make a maximum of 2000 per day if it produced only the basic version. The supply of plastic is

sufficient to produce 1500 dolls per day (both A and B combined). The deluxe version requires a fancy

dress of which there are 600 per day available. If the company makes profit of Rs. 3.00 and Rs. 5.00, per

doll respectively on doll A and B. Formulate this problem as LPP in order to maximize the profit? [06]

__________X__________




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(Session: 2012-13)


Roll No.



[10 Marks]

1) Define O.R. and discuss its characteristics and shortcomings. (Q. No. 2) [02+02]

2) Use Big-M method to maximize: 213 xxz −= s. t. 22 21 ≥+ xx ; 33 21 ≤+ xx ; 42 ≤x and 0, 21 ≥xx .

(Q. No.7) [06]

Part – B

[20 Marks]




Min. 21 25 xxZ −= s. t. 132 21 ≥+ xx ; 0, 21 ≥xx .







__________X__________




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(Session: 2012-13)


Roll No.



[10 Marks]

1) Explain slack and surplus variables with example used in LPP. (Q. No. 4-i) [02+02]

2) Solve the following LPP using simplex method

10834 321 ≤++− xxx

0,, 321 ≥xxxand (Q. No.5) [06]

Part – B

[20 Marks]

3) Solve the following linear programming problem: [08]

Min Z = x1 - 2x2 - 3x3 subject to -2x1 + x2 + 3x3 = 2; 2x1 + 3x2 + 4x3 = 1 and x1, x2, x3 ≥ 0.


Min. 21 25 xxZ −= s. t. 132 21 ≥+ xx ; 0, 21 ≥xx .







__________X__________



42;723.23 21321321 +−≤+−+−= xxxxxtsxxxzMin


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(Session: 2012-13)


Roll No.



[10 Marks]

1) Solve the linear programming problem graphically:

Maximize 21 45 xxz += s. t. 0,;705.12;805.25.1 212121 ≥≤+≤+ xxxxxx . (Q. No. 4-ii) [04]

2) Use Big-M method to maximize: 213 xxz −= s. t. 22 21 ≥+ xx ; 33 21 ≤+ xx ; 42 ≤x and 0, 21 ≥xx .

(Q. No.7) [06]




4) What is operations research? Explain general method for solving operations research models. [02+04]







__________X__________




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(Session: 2012-13)


Roll No.



[10 Marks] 1) Write the dual of the following primal problem [05]

Min z = 3x1 - 2x2 + x3 s.t. 2x1 - 3x2 + x3 ≤ 5; 4x1 - 2x2 ≥ 9; -8x1 + 4x2 + 3x3 = 8 and x1, x2 ≥ 0,

x3 is unrestricted in sign (Q. No. 9-ii T1)

2) A college department chairman has the problem of providing teachers for all courses offered by his

department at the highest possible level of educational 'quality'. He has one professor, two associate

professors, and one teaching assistant (TA) available. Four courses must be offered and, after

appropriate introspection and evaluation he has arrived at the following relative ratings (100 = basic

rating) regarding the ability of each instructor to teach the four courses, respectively. [05]

Courses

1 2 3 4

Prof. 1 60 40 60 70

Prof. 2 20 60 50 70

Prof. 3 20 30 40 60

T. A. 30 10 30 40

How should he assign his staff to the courses to maximize educational quality in his department?

` (Q. No.7 T2)


3) Use dual simplex method to solve the following LPP [7.5]

.0,,,;852;1065;124365..5376

43214321

43243214321

≥≥+++≥−+≥+−++++=

xxxxxxxxxxxxxxxtsxxxxzMin




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4) A company has three plants at locations A, B and C which supply to warehouses located at D, E, F, G

and H. Monthly plant capacities are 800, 500 and 900 units respectively. Monthly warehouse

requirements are 400, 400, 500, 400 and 800 units respectively. Unit transportation costs (in Rupees) are

given below. [7.5]

To

D E F G H

A 5 8 6 6 3

B 4 7 7 6 5

C 8 4 6 6 4

Determine an optimum distribution for the company in order to minimize the total transportation

cost.

5) Formulate the following assignment problem as a linear programming model. [05]

A B C D I 5 7 11 6 II 8 5 9 6 III 4 7 10 7 IV 10 4 8 3

__________X__________


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(Session: 2012-13)


Roll No.



[10 Marks] 1) Consider the following transportation problem: [02+02+02]

It is not possible to transport any quantity from factory B to Godown 5.

Determine: (i) Initial solution by Vogel’s Approximation method.

(ii) Optimum basic feasible solution.

(iii) Is the optimum solution unique? If not, find the alternative

optimum basic feasible solution. (Q. No. 3 T2)

2) There are 5 jobs to be assigned, one each to 5 persons and the profit for each person on each job is

given as: Person [04]

Job

Find the optimal assignment to maximize the profit. (Q. No. 5 T2)


Factory Godowns Stock

Available I II III IV V VI

A 7 5 7 7 5 3 60

B 9 11 6 11 ─ 5 20

C 11 10 6 2 2 8 90

D 9 10 9 6 9 12 50

Demand 60 20 40 20 40 40

A B C D E

I 11 17 8 16 20

II 9 7 12 6 15

III 13 16 15 12 16

IV 21 24 17 28 26

V 14 10 12 11 15



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.0,,,;852;1065;124365..5376

43214321

43243214321

≥≥+++≥−+≥+−++++=





given below. [7.5]

To

D E F G H

A 5 8 6 6 3

B 4 7 7 6 5

C 8 4 6 6 4

Determine an optimum distribution for the company in order to minimize the total transportation cost.

5) Write the dual of the following LPP [05]

.0,;10834;1242;723..23 2132121321321 ≥=++−≥−≤+−−−= xxxxxxxxxxtsxxxzMin

__________X__________


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(Session: 2012-13)


Roll No.



[10 Marks]

1) Use dual simplex method to solve [06]

0,;3;102;7;123 21221212121 ≥≤≥+≤+≥+−−= xxxxxxxxxtosubjectxxzMax (Q. No.10 T1)

2) Write the various steps involved in MODI method. (Q. No.1 T2) [04]

Part – B

[20 Marks]

3) A company has one surplus truck in each of the cities A, B, C, D and E and one deficit truck in each of

the cities 1, 2, 3, 4, 5 and 6. The distance between the cities in kilometers is shown in the matrix below.

Find the assignment of trucks from cities in surplus to cities in deficit so that the total distance covered

by vehicles is minimum. [7.5]

1 2 3 4 5 6

A

B

C

D

E


12 10 15 22 18 8

10 18 25 15 16 12

11 10 3 8 5 9

6 14 10 13 13 12

8 12 11 7 13 10



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given below. [7.5]

To

D E F G H

A 5 8 6 6 3

B 4 7 7 6 5

C 8 4 6 6 4




__________X__________


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(Session: 2012-13)


Roll No.



[10 Marks]

1) A company has factories at F1, F2 and F3 which supply to warehouses at W1, W2 and W3. Weekly

company capacities are 200, 160 and 90 units, respectively. Weekly warehouse requirement are 180,120

and 150 units, respectively. Units shipping costs (in Rupees) are as follows: [05]

Determine the optimal distribution for this company to minimize total shipping cost. (Q. No. 4 T2)

2) A department has five employers with five jobs to be performed. The time (in hours) each men will take

to each job is given in the effective matrix. How should the jobs be allocated, one per employer, so as to

minimize the total man hour? [05]

Jobs Employers

E1 E2 E3 E4 E5 A 10 5 13 15 16 B 3 9 18 13 6 C 10 7 2 2 2 D 7 11 9 7 12 E 7 9 10 4 12

(Q. No. 6 T2)


W W W Supply F 16 20 12 200 F 14 8 18 160 F 26 24 16 90 Demand 180 120 150 450



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Part – B

[20 Marks]


.0,,,;852;1065;124365..5376

43214321

43243214321

≥≥+++≥−+≥+−++++=






1 2 3 4 5 6

A

B

C

D

E



__________X__________

12 10 15 22 18 8

10 18 25 15 16 12

11 10 3 8 5 9

6 14 10 13 13 12

8 12 11 7 13 10


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(Session: 2012-13)


Roll No.



[10 Marks]

1) Write the dual of the following primal problem: [05]

.,0;10;82;22..24. 2121212121 signinedunrestrictxxxxxxxxtsxxzMax ≥≤−=+≥−+=

(Q. No. 9-i T1)

2) Discuss the Hungarian method of solving an assignment problem for minimization. [05]

(Q. No.2 T2)






1 2 3 4 5 6

A

B

C

D

E


12 10 15 22 18 8

10 18 25 15 16 12

11 10 3 8 5 9

6 14 10 13 13 12

8 12 11 7 13 10



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given below. [7.5]

To

D E F G H

A 5 8 6 6 3

B 4 7 7 6 5

C 8 4 6 6 4


5) Use dual simplex method to solve the following LPP [05]

0,1;22 21212121 ≥≥−−≥++= xxandxxxxtosubjectxxzMin

__________X__________


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(Session: 2012-13)


Roll No.



[10 Marks] 1) A small project is composed of seven activities whose time estimates are listed in the table as [05]

follows: Activity

i-j Estimated Duration (in weeks)

Optimistic Most likely Pessimistic 1-2 1 1 7 1-3 1 4 7 1-4 2 2 8 2-5 1 1 1 3-5 2 5 14 4-6 2 5 8 5-6 3 6 15

i) Draw the project network.

ii) Find the expected duration and variance of each activity.

iii) What is the expected project length?

iv) Find critical path.

v) Calculate the variance and standard deviation of project length. (Q. No. 6 T3)

2) What do you understand by sequencing problem? Give three different examples of sequencing problems

of our daily life. (Q. No. 8 T2) [05]


3) The utility data for a network are given below. Determine the total, free, independent and interfering

floats. Identify the critical path and project duration. [06]

Activity 0-1 1-2 1-3 2-4 2-5 3-4 3-6 4-7 5-7 6-7 Duration 2 8 10 6 3 3 7 5 2 8

P. T. O.




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4) A small project consisting of eight activities has the following characteristics: [08]

Activity A B C D E F G H

Preceding Activity - - A A A B, C D E, F, G

Time

Estimates

(in weeks)

Optimistic

time (to)

2 10 8 10 7 9 3 5

Most likely

time(tm)

4 12 9 15 7.5 9 3.5 5

Pessimistic

time(tp)

12 26 10 20 11 9 7 5

(a) Draw the PERT network for the project.

(b) Determine the critical path and the expected project length.

(c) Calculate the variance and standard deviation of the project length.

(d) If a 30 weeks deadline is imposed, what is the probability that the project will be finished in time?

(e) If the project manager wants to be 99%, sure that the project is completed on the schedule date, how

many weeks before that data should he start the project work?

5) Find the sequence that minimizes the total elapsed time required to complete the following tasks, also

find elapsed time and idle time for both machine. [06]

.

__________X__________

Tasks A B C D E F G H I

Time on machine-I 2 5 4 9 6 8 7 5 4

Time on machine-II 6 8 7 4 3 9 3 8 11


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(Session: 2012-13)


Roll No.



[10 Marks] 1) Tasks A,B,C,…..H,I constitute a project. The notation X<Y means that the task X must be finished

before Y can begin. With this notation A<D, A<E, B<F, D<F, C<G, C<H, F<I, G<I [05]

Draw a graph to represent the sequence of tasks and find minimum time of completion of the project,

when the time (in days) of completion of each task is as follows:

Task: A B C D E F G H I

Time: 8 10 8 10 16 17 18 14 9 (Q. No. 3 T3)

2) We have five jobs, each of which must go through the machines A, B and C in the order ABC.

Determine the sequence that will minimize the total elapsed time. [05]

(Q. No. 10 T2)




P. T. O.


Job No.→ 1 2 3 4 5

Machine A 5 7 6 9 5

Machine B 2 1 4 5 3

Machine C 3 7 5 6 7






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Time

Estimates

(in weeks)

Optimistic

time (to)

2 10 8 10 7 9 3 5

Most likely

time(tm)

4 12 9 15 7.5 9 3.5 5

Pessimistic

time(tp)

12 26 10 20 11 9 7 5







5) The utility data for a network are given below. Determine the total, free, independent and interfering floats. Identify the critical path and project duration. [06]

Activity 0-1 1-2 1-3 2-4 2-5 3-4 3-6 4-7 5-7 6-7 Duration 2 8 10 6 3 3 7 5 2 8

__________X__________


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(Session: 2012-13)


Roll No.



[10 Marks]

1) Explain the following terms used in PERT: (i) Pessimistic time (ii) Optimistic time (iii) Most likely

time. (Q. No. 1 T3) [05]

2) There are six jobs, first over machine I and then over II. The order of the completion of jobs has no

significance. The following table gives the machine times in hours for six jobs and the two machines.

Job No.→ 1 2 3 4 5 6 Machine I 5 9 4 7 8 6 Machine II 7 4 8 3 9 5

Find the sequence of the jobs that minimizes the total elapsed time to complete the jobs. Find the minimum time also. (Q. No. 9 T2) [05]

Part – B

[20 Marks]

3) The utility data for a network are given below. Determine the total, free, independent and interfering floats. Identify the critical path and project duration. [06]

Activity 0-1 1-2 1-3 2-4 2-5 3-4 3-6 4-7 5-7 6-7

Duration 2 8 10 6 3 3 7 5 2 8

P. T. O.




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Time

Estimates

(in weeks)

Optimistic

time (to)

2 10 8 10 7 9 3 5

Most likely

time(tm)

4 12 9 15 7.5 9 3.5 5

Pessimistic

time(tp)

12 26 10 20 11 9 7 5








find elapsed time and idle time for each machine. [06]

__________X__________

Tasks A B C D E F G

Time on machine-I 3 8 7 4 9 8 7

Time on machine-II 4 3 2 5 1 4 3

Time on machine-III 6 7 5 11 5 6 12


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(Session: 2012-13)


Roll No.



[10 Marks]

1) A project has the following time schedule: [05]

Activity: 1-2 1-3 1-4 2-5 3-6 3-7 4-6 5-8 6-9 7-8 8-9

Duration: 2 2 1 4 8 5 3 1 5 4 3 (in months) Construct PERT network and compute

(i) Total float, free float and independent float for each activity,

(ii) Critical path and its duration. (Q. No. 8 T3)

2) Discuss in brief PERT, CPM and dummy activities with reference to Project Management. [05]

(Q. No. 2 T3)

Part – B

[20 Marks]



P. T. O.







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Time

Estimates

(in weeks)

Optimistic

time (to)

2 10 8 10 7 9 3 5

Most likely

time(tm)

4 12 9 15 7.5 9 3.5 5

Pessimistic

time(tp)

12 26 10 20 11 9 7 5









__________X__________

Tasks A B C D E F G





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(Session: 2012-13)


Roll No.



[10 Marks]

1) A project has the following characteristics: [05]

Activity Preceding activity Expected completion time A None 5 B A 2 C A 6 D B 12 E D 10 F D 9 G D 5 H B 9 I C, E 1 J G 2 K F, I, J 3 L K 9 M H, G 7 N M 8

(i) Draw a PERT network for this project. (ii) Find the critical path and project completion time.

(iii) Calculate total float for each activity. (Q. No. 4 T3)

2) The following are the details of estimated times of activities of a certain project. [05]

Activity A B C D E F Immediate predecessors

- A A B,C - E

Estimated time(weeks)

2 3 4 6 2 8

P. T. O.




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(a) Find the critical path and the expected time of the project. (b) Calculate the earliest start time and earliest finish time for each activity. (c) Calculate the slack for each activity. (Q. No. 11 T3)







Time

Estimates

(in weeks)

Optimistic

time (to)

2 10 8 10 7 9 3 5

Most likely

time(tm)

4 12 9 15 7.5 9 3.5 5

Pessimistic

time(tp)

12 26 10 20 11 9 7 5









__________X__________




Tasks A B C D E F G





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(Session: 2012-13)


Roll No.



[10 Marks] 1) In a game of matching coins with two players, suppose A wins one unit of value, when there are two

heads, wins nothing when there are two tails and losses 21 units of value when there are one head and

one tail. Determine the payoff matrix, the best strategies for each player and the value of the game to A.

(Q. No. 3) [05]

2) A TV repairman finds that the time spent on his jobs has an exponential distribution with mean 30

minutes. If he repairs sets in the order in which they came in, and if the arrival of sets is approximately

Poisson with an average rate of 10 per 8 hour day, what is repairman’s expected idle time each day?

How many jobs are ahead of the average set just brought? (Q. No. 10) [05]


3) In a certain game player has three possible courses of action L, M and N, while B has two possible

choices P and Q. Payments to be made according to the choice made. [06]

Choices Payments L,P A pays B Rs.3 L,Q B pays A Rs. 3 M,P A pays B Rs.2 M,Q B pays A Rs.4 N,P B pays A Rs.2 N,Q B pays A Rs.3

What are the best strategies for players A and B in this game? What is the value of the game for A and

B?

4) On an average 96 patients per 24-hour day require the service of an emergency clinic. Also on an

average, a patient requires 10 minutes of active attention. Assume that the facility can handle only one

emergency at a time. Suppose that it costs the clinic Rs. 100/- per patient treated to obtain an average




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servicing time of 10 minutes, and that each minute of decrease in this average time would costs Rs. 10/-

per patient treated. How much would have to be budgeted by the clinic to decrease the average size of

the queue from 311 patients to

21 a patient. [06]

5) For the following payoff matrix, transform the zero-sum game into an equivalent linear programming

problem and solve it by using simplex method. [08]

Player B

Player A

__________X__________

B1 B2 B3 A1 1 -1 3 A2 3 5 -3 A3 6 2 -2


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(Session: 2012-13)


Roll No.



[10 Marks] 1) A and B play game in which each has three coins, a 5p., 10p. and a 20p. Each selects a coin without the

knowledge of the other’s choice. If the sum of the coins is an odd amounts, A wins B’s coin, if the sum is even, B

wins A’s coin. Find the best strategy for each player and the value of the game. (Q. No. 4) [05]

2) A telephone exchange has two long distance operators. The telephone company finds that, during the

peak load, long distance calls arrive in Poisson fashion at an average rate of 15 per hrs. The length of the

service on these calls is approximately exponentially distributed with mean length of 5 minutes.

(a) What is the probability that a subscriber will have to wait for his long distance call during the peak

hrs of the day? (b) If the subscribers will wait and serviced in turn, what is the expected waiting time?

(Q. No. 9) [05]





What are the best strategies for players A and B in this game? What is the value of the game for A and B?







Page-2 of 2




21 a patient. [06]



Player B

Player A

__________X__________

B1 B2 B3 A1 1 -1 3 A2 3 5 -3 A3 6 2 -2


Page-1 of 2



(Session: 2012-13)


Roll No.



[10 Marks] 1) Solve the following games graphically.

Player B

Player A

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

1234976521

(Q. No. 5-ii) [05]

2) If for a period of two hrs. in the day (8-10 A.M.), trains arrive at the yard every 20 minutes but the

service time continuous to remain 36 minutes. Then calculate for this period (a) the probability that yard

is empty. (b) Average queue length, on the assumption that the line capacity of the yard is limited to 4

trains only. (Q. No. 8) [05]





What are the best strategies for players A and B in this game? What is the value of the game for A and

B?




Page-2 of 2







21 a patient. [06]



Player B

Player A

__________X__________

B1 B2 B3 A1 1 -1 3 A2 3 5 -3 A3 6 2 -2


Page-1 of 2



(Session: 2012-13)


Roll No.



[10 Marks]

1) Consider a “modified” form of “matching biased coins” game problem. The matching player is paid Rs.

8.00 if the two coins turn both heads and Rs.1.00 if the coins turn both tails. The non-matching player is

paid Rs.3.00 when the two coins do not match. Given the choice of being the matching or non-matching

player, which one would you choose and what would be your strategy? (Q. No. 6) [05]

2) A self-service store employs one cashier at its counter. An average of nine customers arrives every 5

minutes while the cashier can serve 10 customers in 5 minutes. Assuming Poisson distribution for arrival

rate an exponential distribution for service rate, find (i) average number of customer in the system (ii)

average number of customer in queue or average queue length (iii) average time a customer spends in

the system (iv) average time a customer waits before being served. (Q. No. 11) [05]

Part – B

[20 Marks]










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21 a patient. [06]



Player B

Player A

__________X__________

B1 B2 B3 A1 1 -1 3 A2 3 5 -3 A3 6 2 -2


Page-1 of 2



(Session: 2012-13)


Roll No.



[10 Marks]

1) Solve the following games graphically.

Player B

Player A ⎥⎦

⎤⎢⎣

⎡ −23012012

(Q. No. 5-i) [05]

2) Regarding queue system, explain the following: (i) Balking, reneging & jockeying, (ii) steady state &

transient state. (Q. No. 7-iv & vi) [05]

Part – B

[20 Marks] 3) In a certain game player has three possible courses of action L, M and N, while B has two possible










21 a patient. [06]




Page-2 of 2



Player B

Player A

__________X__________

B1 B2 B3 A1 1 -1 3 A2 3 5 -3 A3 6 2 -2

b. tech. [sem vii (ce, me, it, ei, en, cs & ec – all...

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